1. Remove the plagiarism to below 15% without changing the science. You need a good knowledge of thermoelectric properties of a material. Do NOT use a software.2. Correct grammatical mistakes.

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Sue25

ORIGINALITY REPORT

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SIMILARIT Y INDEX

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INT ERNET SOURCES

PUBLICAT IONS

ST UDENT PAPERS

PRIMARY SOURCES

1

Jing Wu, Yabin Chen, Junqiao Wu, Kedar

Hippalgaonkar. “Perspectives on

Thermoelectricity in Layered and 2D Materials”,

Advanced Electronic Materials, 2018

24%

Publicat ion

2

3

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5

6

www.ideals.illinois.edu

Int ernet Source

www.coursehero.com

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export.arxiv.org

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www.shobhituniversity.ac.in

Int ernet Source

Submitted to Malaviya National Institute of

Technology

13%

8%

2%

2%

2%

St udent Paper

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Oshman, Christopher, Charles Opoku,

Abhishek S. Dahiya, Daniel Alquier, Nicolas

Camara, and Guylaine Poulin-Vittrant.

1%

“Measurement of Spurious Voltages in ZnO

Piezoelectric Nanogenerators”, Journal of

Microelectromechanical Systems, 2016.

Publicat ion

8

Karamitaheri, Hossein, Mahdi Pourfath, Rahim

Faez, and Hans Kosina. “Atomistic Study of the

Lattice Thermal Conductivity of Rough

Graphene Nanoribbons”, IEEE Transactions on

Electron Devices, 2013.

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digbib.ubka.uni-karlsruhe.de

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www.nmletters.org

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www.niscair.res.in

Int ernet Source

B. Iniguez. “Compact-Modeling Solutions For

Nanoscale Double-Gate and Gate-All-Around

MOSFETs”, IEEE Transactions on Electron

Devices, 9/2006

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Submitted to University of Florida

St udent Paper

www.nature.com

Int ernet Source

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B. Govoreanu. “On the calculation of the quasibound-state energies and lifetimes in inverted

MOS structures with ultrathin oxides and its

application to the direct tunneling current”,

IEEE Transactions on Electron Devices, 5/2004

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searchlibrary.ohchr.org

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d-nb.info

Int ernet Source

Sébastien Glaser, Saïd Mammar, Chouki

Sentouh. “Integrated Driver–Vehicle–

Infrastructure Road Departure Warning Unit”,

IEEE Transactions on Vehicular Technology,

2010

1%

Publicat ion

19

Milo Yaro Swinkels, Ilaria Zardo. “Nanowires for

heat conversion”, Journal of Physics D: Applied

Physics, 2018

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Sue25
GRADEMARK REPORT
FINAL GRADE
GENERAL COMMENTS
/0
Instructor
PAGE 1
PAGE 2
PAGE 3
PAGE 4
PAGE 5
PAGE 6
1.1 Bardeen transfer Hamiltonian approach
Model electron transport in graphene-hBN tunnel transistors – months 6-12: I will combine a
model of the electrostatics of the graphene-hBN tunnel device with the Bardeen transfer Hamiltonian
approach to determine the transmission coefficient of electrons tunnelling between closely aligned
graphene lattices.
Separability of System
Figure 1.
[1]
Separate total system into distinct subsystems with known Hamiltonians, wave solutions
Full, exact Hamiltonian, H
𝐻 = 𝐻𝐿 + 𝐻𝑅 + 𝐻𝑇
𝐻𝐿 + 𝐻𝑅 known while 𝐻𝑇 unknown. Transfer Hamiltonian[1]
ℏ2
𝐻𝐿 = − 2𝑚 ∇2 + 𝑉𝐿 (𝑟⃑)
ℏ2
𝐻𝑅 = − 2𝑚 ∇2 + 𝑉𝐿 (𝑟⃑)
𝑟⃑ ∈ 𝑅𝐿
𝑟⃑ ∈ 𝑅𝑅
Fermi’s Golden Rule and the Matrix Element
Probability of Elastic Tunneling Probability of Transition from to :
2𝜋
𝑃 = ( ℏ ) ∑𝑣|𝑀|2 𝛿(𝐸𝑅,𝑣 − 𝐸𝐿,0 ) , 𝑀 = ⟨𝜒𝑣 |𝐻′|𝜑0 ⟩
(𝐻 ′ = 𝐻𝑅 + 𝐻𝑇 )
But, how can we find the tunneling matrix element, M ?[2]
Substitute for 𝐻′:
𝐻 ′ = 𝐻 − 𝐻𝐿
∞
𝑀 = ∫ 𝜒𝑣∗ (𝐻 − 𝐻𝐿 )𝜑0 𝑑𝑟⃑
−∞
We don’t know the full H, but can we find a suitable approximation for it ?
The Model Hamiltonian
Bardeen proposed a model Hamiltonian:
𝐻
𝐻~𝐻𝑀 = { 𝐿,
𝐻𝑅
𝑟⃑ ∈ 𝑅𝐿
𝑟⃑ ∈ 𝑅𝑅
Which will greatly simplify calculation of !
Remember the separable subsystems figure above,
𝜑𝑖 (𝑟⃑) decays across the barrier and is ~ 0 in 𝑅𝑅 , meaning in that region, 𝐻~𝐻𝑅
Similarly for 𝜒𝑗 (𝑟⃑), so 𝐻~𝐻𝐿, in 𝑅𝐿
1.1.1 The Matrix Element, M
[1]
`
ℏ2
𝑀=
∫ 𝜑0 ∇𝜒𝑣∗ − 𝜒𝑣∗ ∇𝜑0 𝑑𝑆
2𝑚
𝑆𝐿𝑅
𝑆𝐿𝑅 is a surface separating 𝑅𝐿 and 𝑅𝑅
Calculation of 𝑀 requires knowledge only of wavefunctions 𝜑0 (𝑟⃑) and 𝜑𝑣 (𝑟⃑) of individual systems,
not full system Hamiltonian or wave function!
Can easily calculate transmission coefficient,
2𝜋
𝑇~𝑃 = ( ) ∑|𝑀|2 𝛿(𝐸𝑅,𝑣 − 𝐸𝐿,0 )
ℏ
𝑣
1.2 Landauer approach
Optimizing the Seebeck coefficient –months 12-18: Using (2) and the Landauer approach. I will
calculate the Seebeck coefficient, S, of the tunnel transistors, the figure of merit for thermoelectric
effects. I will use my model to optimize S by considering the electrostatic configuration and the
lattice alignment.
1.2.1 Landauer Formula
The quantized conductance through the channel without scattering:
𝑮𝟎 =
𝟐𝒆𝟐
1
𝒉
the overall conductance with scattering effect is given by the Landauer formula [3] which is the quantum of conductance times
the probability of electron transmission through the channel.
𝟐𝒆𝟐
̅̅̅̅
𝐆=
𝑻
2
𝒆𝒍 (𝑬)
𝒉
This equation is called the Landauer formula.
The net current should take into account finite temperature or voltage differences applied to the wire
by multiplying the variation of Fermi-Dirac distributions of the electrons in the left and the right,
integrated over all energies[4].
𝑰(𝑬) =
𝟐𝒆
𝒉
+∞
̅̅̅̅
∫−∞ 〖𝑻
𝒆𝒍 (𝑬)[ 〗𝒇𝒔 (𝑬) − 𝒇𝒅 (𝑬)]𝒅𝑬
3
Here, 𝑓𝑠 (𝐸) and 𝑓𝑑 (𝐸) are the Fermi distribution functions of the source and drain contacts,
respectively.
When both voltage and temperature differences are applied to the system, a difference of the Fermi
distributions over all the energy levels can be simply expressed by the superposition sum of each of
the two cases.
𝒇𝒔 (𝑬) − 𝒇𝒅 (𝑬) ≈ (−
𝛛𝐟𝟎
𝛛𝐄
) 𝐪∆𝐕 − (−
𝛛𝐟𝟎 𝐄−𝐄𝐅
𝛛𝐄
)
𝐓
∆𝐓
4
Thermoelectric properties can be evaluated using the Landauer approach. Current equations with the
Landauer formalism in the linear response regime can be expressed as a combination of electric
potential and temperature contributions with electrical transport properties as shown below[4].
̅̅̅̅(𝐄)∆𝑻
𝑰(𝑬) = 𝑮(𝑬)∆𝑽 + 𝑺𝑮
5
̅̅̅̅(𝑬)∆𝑽 − 𝑲𝒐 ∆𝑻
𝑰𝒒 (𝑬) = −𝑻𝑺𝑮
6
where 𝐼 and 𝐼𝑞 are the electric and the heat current, respectively. Here, 𝐾𝑜 is the electronic
contribution to the thermal conductivity for zero electric field, defined as[5]
Plugging in Eq. 4 into Eq. 3 gives the same form as Eq. 5 with the voltage and temperature terms,
and a comparison of these equations defines the electrical conductance 𝐺, and the thermal
conductance 𝐾𝑜 for zero electric current as [5]:
𝑮(𝑬) =
𝑲𝒐 =
𝟐𝒆
𝒉
+∞
𝛛𝐟
𝟎
̅̅̅̅
∫−∞ 𝑻
𝒆𝒍 (𝑬) (− 𝛛𝐄 ) 𝒅𝑬
𝟐 +∞
̅̅̅̅(𝑬)(𝑬
∫ 𝑻
𝒉𝑻 −∞ 𝒆𝒍
− 𝑬𝑭 )𝟐 (−
7
𝝏𝒇
𝝏𝑬
) 𝒅𝑬
8
The Seebeck coefficient can be evaluated by 𝑆 = ̅̅̅̅
𝑆𝐺 /𝐺 as [5]:
𝑺(𝑬) =
+∞
𝝏𝒇
̅̅̅̅
𝒆𝒍 (𝑬)(𝑬−𝑬𝑭 )(−𝝏𝑬)𝒅𝑬
𝟏 ∫−∞ 𝑻
−𝒆𝑻
9
+∞
𝝏𝒇
𝑻𝒆𝒍 (𝑬)(− )𝒅𝑬
∫−∞ ̅̅̅̅
𝝏𝑬
1.3 Nonequilibrium Green’s function
Electron-phonon interaction and evaluation of different heterostructure materials investigate
electron-phonon scattering via nonequilibrium Green’s function approach and the effect on S of
using different contact and barrier materials
1.3.1 Nonequilibrium Green’s function
1.4 Fabricate and measure devices
1.4.1 Thermoelectric measurement of 2D materials.
Figure 2. Thermoelectric measurement of 2D materials
In this study, we shall use a setup as shown in figure 2. Due to the nanoscale nature of
layered 2D materials, the thermoelectric measurement techniques are different from those employed
for bulk materials. The main challenge to measuring the thermoelectric power- factor (S2σ) is to
obtain the Seebeck coefficient S = −V/ΔT, which is defined as the ratio of open-circuit voltage V to
the temperature difference ΔT along the 2D materials. The direct measurement of this Seebeck
coefficient in 2D materials is possible by using locally fabricated microresistance thermometers
across the 2D material as shown in Figure 2[6]. Such a technique has been used for many 2D
materials including graphene [7]–[9] as well as 1D nanowires[10]–[12] and nanotubes[13]. A
temperature gradient is generated along the 2D material by applying a heating current Ih to the
microheater through Joule heating. Then, a thermoelectric voltage V will be generated along this
temperature gradient. The measurement of the Seebeck coefficient is thus divided into two parts: the
measurement for V and ΔT.
There are two different methods that can be used to measure V depending on whether a DC or an AC
heating current is applied to the microheater[6]. By sweeping a DC current in the micro- heater, a
voltage difference V can be measured according to the heating current. A parabolic V–Ih curve is
obtained due to the relation: V ∝ ΔT ∝ Ih2. We can get a very accurate V measurement by this DC
sweeping method. However, this requires a slow sweep of Ih (to ensure thermal equilibrium) and can
be quite time consuming. The AC method, which typically uses a lockin amplifier to direct lock the
V signal frequency induced by the heating current requires signifi- cantly less time. An AC heating
current Ih = Isin(ωt) with the frequency of ω is applied. As the thermoelectric voltage is proportional
to the temperature difference along the 2D material V ∝ ΔT ∝ Ih2 = I2sin2(ωt), one can hence
measure the thermoelectric voltage by sensing the 2ω signal, which is π/2 out of phase with the
heater signal. However, for high resistance 2D materials, this measurement is limited by the input
impedance of the lockin amplifier. After the measurement of ΔT and V, we can now extract the
Seebeck coefficient of the measured 2D materials by S = −V/ΔT. Employing such techniques, we
will measure the in-plane thermoelectric properties of our 2D materials heterostructrure.
1.5 Electron-phonon interaction and evaluation of different heterostructure materials
2
Data and data analysis
3
Conclusion
These results will be expected to suggest prospective ideas to improve the current thermoelectric
technology.
[1]
P. Albrecht, K. Ritter, and L. Ruppalt, “The Bardeen Transfer Hamiltonian Approach to
Tunneling and its Application to STM and Carbon Nanotubes,” 2004.
[2]
J. Bardeen, “Recursos sobre protección de los derechos de la mujer en internet (conté
‘disquet’),” Phys. Rev. Lett., vol. 6, no. 2, pp. 57–59, 1961.
[3]
S. Datta, Quantum Transport : Atom to Transistor. Cambridge University Press, 2005.
[4]
K. H. Park, “Theoretical Investigation of Thermoelectric Properties of,” 2012.
[5]
C. Jeong, R. Kim, M. Luisier, S. Datta, and M. Lundstrom, “On Landauer versus Boltzmann
and full band versus effective mass evaluation of thermoelectric transport coefficients,” J.
Appl. Phys., vol. 107, no. 2, 2010.
[6]
J. Wu, Y. Chen, J. Wu, and K. Hippalgaonkar, “Perspectives on Thermoelectricity in Layered
and 2D Materials,” vol. 1800248, pp. 1–18, 2018.
[7]
M. Yoshida et al., “Gate-Optimized Thermoelectric Power Factor in Ultrathin WSe2Single
Crystals,” Nano Lett., vol. 16, no. 3, pp. 2061–2065, 2016.
[8]
P. Wei, W. Bao, Y. Pu, C. N. Lau, and J. Shi, “Anomalous thermoelectric transport of dirac
particles in graphene,” Phys. Rev. Lett., vol. 102, no. 16, pp. 1–4, 2009.
[9]
Y. M. Zuev, W. Chang, and P. Kim, “Thermoelectric and magnetothermoelectric transport
measurements of graphene,” Phys. Rev. Lett., vol. 102, no. 9, pp. 1–4, 2009.
[10] M. Cimen, “The effect of separation from dam on suckling duration and frequency of lambs,”
Indian J. Anim. Res., vol. 46, no. 3, pp. 284–287, 2012.
[11] X. P. a Gao et al., “One-Dimensional Quantum Confinement Effect Modulated
Thermoelectric Properties in InAs Nanowires.,” Nano Lett., 2012.
[12] S. Roddaro et al., “Giant thermovoltage in single InAs nanowire field-effect transistors,” Nano
Lett., vol. 13, no. 8, pp. 3638–3642, 2013.
[13] C. Yu, L. Shi, Y. Zao, D. Li, and A. Majumdar, “Thermal Conductance of an Individual
Single-Wall Carbon Nanotube above Room Temperature,” Nano Lett., vol. 6, no. 1, pp. 96–
100, 2005.
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